The main evidence to support this claim comes from the actions of the Japanese as the fighting got nearer to the Japanese homeland. The nearer it came, the more ferocious it got, and the preparations the Japanese were making to resist an American invasion of the home islands make it clear that such an invasion would have caused huge numbers of casualties.
On the islands of Saipan, Iwo Jima, and Okinawa in particular, the Japanese fought almost to the last man. On Saipan, even civilians committed suicide rather than be captured. In the battle for Okinawa, the Japanese started to use large numbers of kamikaze attacks against the American fleet. These sorts of actions show the Japanese were not going to give up the fight.
This is also shown by the plans the Japanese were making to resist an American invasion. These plans involved many different kinds of suicide attacks on the invaders. They also involved using Japanese civilians as combatants. Such an invasion would have been horrendously bloody. These sorts of preparations, combined with evidence from the fighting, can be used to prove that the Japanese would not have surrendered to any conventional pressures.